Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#29144 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C43×C53 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,6)(3,5,7,8)(9,10,13,16)(11,15,12,14)(17,18,22,30,20,23,21,25)(19,26,29,24,27,32,28,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,7)(2,5,6,8)(9,11,13,12)(10,14,16,15)(18,23)(24,31)(25,30)(26,32);(3,7)(5,8)(17,19,20,27)(18,24,23,31)(21,29,22,28)(25,32,30,26);(9,12,13,11)(10,15,16,14)(18,23)(24,31)(25,30)(26,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 22132);